The lifespans of turtles in a particular zoo are normally distributed. The average turtle lives $106$ years; the standard deviation is $19$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a turtle living between $68$ and $87$ years.
$106$ $87$ $125$ $68$ $144$ $49$ $163$ $95\%$ $68\%$ $13.5\%$ $13.5\%$ We know the lifespans are normally distributed with an average lifespan of $106$ years. We know the standard deviation is $19$ years, so one standard deviation below the mean is $87$ years and one standard deviation above the mean is $125$ years. Two standard deviations below the mean is $68$ years and two standard deviations above the mean is $144$ years. Three standard deviations below the mean is $49$ years and three standard deviations above the mean is $163$ years. We are interested in the probability of a turtle living between $68$ and $87$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the turtles will have lifespans within 2 standard deviations of the average lifespan. It also tells us that $68\%$ of the turtles will have lifespans within 1 standard deviation of the mean. The probability of a particular turtle living between $68$ and $87$ years is $\color{orange}{13.5\%}$.